The present invention pertains to the art of contrast phantoms used to calibrate active millimeter wave imaging systems. Millimeter wave imaging is employed in personnel screening systems to detect concealed explosives and weapons. In order to benchmark performance of an active millimeter wave system, it is desirable to have a phantom that contains materials that reflect incoming radiation such that an analysis can be performed on the system to verify results. The intensity of millimeter radiation observed from illumination of a target material depends on the values of the real and imaginary components of the complex dielectric constant, and the target material's geometric thickness. This has been described in “Millimeter Wave Measurements of Explosives and Simulants” by Barber, et al. in Proc. of SPIE Vol. 7670, 76700E, incorporated herein by reference. Briefly, the intensity returned from a material is
                    I        =                              [                                          R                F                            +                                                                                          (                                              1                        -                                                  R                          F                                                                    )                                        ⁢                    2                                                        1                    -                                                                  R                        F                                            ⁢                                              R                        R                                            ⁢                                              exp                        ⁡                                                  (                                                                                    -                              2                                                        ⁢                            KL                                                    )                                                                                                                    ⁢                                  R                  R                                ⁢                                  exp                  ⁡                                      (                                                                  -                        2                                            ⁢                      KL                                        )                                                                        ]                    *                      I            0                                              (        1        )            where RF is the front surface reflection, RR the rear surface the reflection, L is the length of the material, and K is the attenuation in the material. The front and rear surface reflections are functions of the dielectric constant of the material itself and the materials at the interfaces. Using air as the surrounding material, it is possible to plot the reflectivity of an arbitrary material of infinite thickness as a function of real and imaginary dielectric constant (∈ and ∈″), yielding a plot 10 as illustrated in FIG. 1 showing Reflectivity R at the air/surface boundary of a material of complex dielectric constant e. Contours 15 of constant R range from 0.1 to 0.7. This demonstrates that for a reflectivity of interest, there is an ellipsoid of solutions for combinations of real and imaginary dielectric values which will yield that particular reflectivity from a material. It is only necessary to tailor the material to match one point on the ellipsoid of solutions.
The propagation of electromagnetic plane waves is well understood. Consider a plane was propagating through free space where the amplitude of the electric field is E0. The dielectric constant of free space (vacuum) is defined as 1−i0, where air can be substituted for vacuum to a first approximation. As the plane wave meets a boundary of a medium at normal incidence, the plane wave is both reflected and transmitted. The amplitude of the reflected and transmitted waves are defined asEr=rE0,  (2)Et=tE0  (3)where r and t are the reflection and transmission coefficients.
                    r        =                                                      1              -                                                ɛ                  matl                                                                    1              +                                                ɛ                  matl                                                                                                  (        4        )                                t        =                              (                          1              -              r                        )                    .                                    (        5        )            Note that r and t are related to the electric field, and are to be distinguished from reflectance R and transmittance T, defined asR=|r|2  (6)T=|t|2  (7)which are related to intensity (the square of the electric field). The complex dielectric constant (relative permittivity) of a material is defined as:∈matl=∈matl′−i∈matl″  (8)where ∈′ and ∈″ are the real and imaginary portions of the complex dielectric constant. The dielectric constant of a material is a frequency-dependent quantity and describes how a material responds to electromagnetic waves. The dielectric constant can be used interchangeably with complex index of refraction as they are related quantities.
A detailed derivation of the reflected and transmitted waves for a plane wave normally incident on a plane-parallel slab of arbitrary material with particles much smaller than the wavelength of the radiation has been described in “Absorption and Scattering of Light by Small Particles” Wiley-VCH Weinheim (2004), incorporated herein by reference. The reflected wave can be derived by considering the dielectric constants of the slab and the surrounding material. For a slab of material of thickness L in air, the reflection coefficient of the slab is given by:
                              r          slab                =                                                      r              ⁡                              [                                  1                  -                                      exp                    ⁡                                          (                                              ⅈ                        ⁢                                                                                                  ⁢                                                                              4                            ⁢                            π                                                    λ                                                ⁢                        L                        ⁢                                                                              ɛ                            slab                                                                                              )                                                                      ]                                                    1              -                                                r                  2                                ⁢                                  exp                  ⁡                                      (                                          ⅈ                      ⁢                                                                                          ⁢                                                                        4                          ⁢                          π                                                λ                                            ⁢                      L                      ⁢                                                                        ɛ                          slab                                                                                      )                                                                                                                    (        9        )            where λ is the wavelength of the incoming radiation. For an imaging portal operating over a narrow band of frequencies, the signal returned is determined by the thickness of the material and the dielectric constant. Thus, in order to simulate the reflected signal expected from a slab of explosives, it is necessary to match the complex dielectric constant of the explosive with a different material to be used as the simulant. In doing so, one can then configure the simulant identically to an explosive and expect the same return. Note that rslab→r when the term in the exponential becomes large through some combination of increasing thickness or complex dielectric constant of the slab. As such, for opaque materials, there is an ellipsoid of solutions for the complex dielectric constant that will produce a desired reflectivity. This greatly simplifies the process of creating simulants, as it is only necessary to find a material whose complex dielectric constant falls on this ellipsoid.
The theory of mixing component materials with regard to dielectric constant is fairly well developed. The Landau & Lifshitz, Looyenga equation, or(∈mixture)1/3=ν1(∈1)1/3+ν2(∈2)1/3  (10)provides a good approximation for determining the dielectric constant of a mixture from those of the mixture's individual component materials. Here, e is the complex dielectric constant and u is the volume fraction of the component material. This has been used as a starting point for creating millimeter wave simulants. Once the dielectric constant of a base has been identified and measured, adjustments to the base material can be made by doping with materials with different dielectric, constants to change both the real and imaginary portions of the formulation. When coupled with the thickness of the material, Equation (10) can be used to create a material of any reflectivity.
As the development of active millimeter wave imaging systems continues, it is necessary to validate materials that simulate the expected response of explosives. Further, while physics-based models have been used to develop simulants, there exists a need in the art to image both the explosive and simulant together in a controlled fashion and in order to benchmark performance of an active millimeter wave system, and there exists a need to have a contrast phantom containing materials that reflect incoming radiation and allow an analysis to be performed to verify the results generated by an active millimeter wave imaging system are correct.